Remark 2.2 
In our setting, for overcoming the circular dependence between the controlled filtration \(\mathcal {F}_t^{Y^u}\)  and the control \(u\) , we have adopted the Girsanov measure transformation method. We point out that our framework is different from that most used in ([1]}, [2]}, [3]}, [4]}, [5]}), but similar as ([6]}, [7]}). The obvious difference between these two kinds of frameworks depends on the original probability measure, that is, \(P\)  or \(\bar{P}\) , which is given first. In the first framework, for a given original probability measure \(P\) , the observation process \(Y\)  is given as a standard Brownian motion. However, in the second framework which is used in our paper, under a given original probability measure \(\bar{P}\) , the observation process \(Y\)  is a stochastic process which becomes a standard Brownian motion under the new probability measure \(P\)  by Girsanov's theorem.
